On an interval \(I\subseteq \mathbb {R}\) , for any \(p\ge 1\)  and any two atomless measures, the optimal transport plan is induced by an optimal transport map [1]}, i.e., \(W_p^p(\mu _1, \mu _2) = K_p(T)\) , with a monotonic increasing \(T\)  defined by
\(T= F_{\mu _1}^{-1} \circ F_{\mu _2} \, ,\) 
